Recently I had the occasion to read a question and answer column by a priest. In the column a woman mentioned that her son was a God parent at the baptism of a child born to a woman in an unmarried state. The inquirer wanted to know if it was proper for such a child to be baptized. The priest replied that of course it was. He did not however address the underlying assumptions of the inquirer; that the Church would withhold baptism from an innocent due to the sins of it's parents.
I find that assumption most disturbing and counter productive. An unwed birth is most often, especially in the case of a teenage mother, the result of a very bad decision. It could only be more terrible for such a mother to make an even worse bad decision and murder her baby through the sin of abortion.
So the right thing for her to do is have her baby. Some would then cut her off from the pastoral guidance of the Church and deny salvation to the innocent child, for some perceived sin of public scandal.
The sin of an unwed mother is quite visible. The sins of the rest of us are not quite so visible but every bit as real. Our sins, just as hers, are forgiven through the sacrament of Reconciliation.
So my assumption would be that the Church would be quite willing to extend the Sacrament of Baptism for the child and Eucharist for the mother to a repentant sinner, just as she does for all the rest of us repentant sinners. And if we pray for her, perhaps she'll also pray for us.